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Quantum scattering

  1. Jun 12, 2009 #1
    1. The problem statement, all variables and given/known data

    A beam of particles, each of mass m and energy E0,is incident on a square potential energy well of width L and depth V0,where 0 <V0 < h(bar)^2[tex]\pi[/tex]^2/2mL^2 . Outside the region of the well, the potential energy is equal to zero. Suppose that E0 is the lowest energy at which a transmission resonance occurs, with 100% of the beam being transmitted and none reflected.

    (a)
    A second beam of particles, each of mass m and energy E1 is incident
    on a square potential energy barrier of width L and height V0. Outside the
    region of the barrier, the potential energy is equal to zero. What is the
    lowest value of E1 at which a transmission resonance occurs in this
    situation, with 100% of the beam being transmitted and none reflected?
    Express your answer in terms of E0 and V0.
    (b)
    A third beam of particles, each of mass m and energy V0/2is
    incident on a square potential energy barrier of width L and height V0.
    Outside the region of the barrier, the potential energy is equal to zero.
    Estimate the fraction of particles that tunnels through the barrier if
    V0 =1.0h(bar)^2/mL^2 .
    (c)
    Suppose that the barrier in part (b) extends from x =0 to x = L and the incident beam travels in the positive x-direction. For x< 0 the energy eigenfunction describing the beam is Aexp(ikx) + Bexp(−ikx), while for x>L it is Fexp(ikx),where k is the wave number and A, B and F are constants. For the conditions described in part (b), what is the value of the ratio |F|/|B|?


    2. Relevant equations



    3. The attempt at a solution

    Can someone give me help with starting this please?

    Is the lowest energy E0 = V0/2 ?

    if so how is this derived?
     
    Last edited: Jun 12, 2009
  2. jcsd
  3. Jun 13, 2009 #2

    drizzle

    User Avatar
    Gold Member

    first you need to sketch the problem, so you can understand it well [I've done it for you:smile:], apparently you have three zones as shown bellow

    in a general matter, you have to solve the Schrödinger equation in the x direction, you should have 3 equations for this system [one for each zone], you also need to consider the primary conditions to solve it.

    hint: with the beam being 100% transmitted, what do you think the energy of the particles in this system should be, E>Vo or E<Vo?
     

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